The MAP kinase cascade is a group of enzymes which can iteratively add phosphate groups to each other. More specifically, when a suitable number of phosphate groups have been added to one enzyme in the cascade it becomes activated and can add a phosphate to the next enzyme in the row. I found this kind of idea of enzymes modifying each other with the main purpose of activating each other fascinating when I first came across it. (The first example I saw was actually the complement cascade which occurs in immunology.) This type of structure is just asking to be modelled mathematically and not surprisingly a lot of work has been done on it. Here I will survey some of what is known.
The MAP kinase cascade is a structure which occurs in many types of cells. It has three layers. The first layer consists of a protein which can be phosphorylated once. The second layer consists of a protein which can be phosphorylated twice by the same enzyme. This enzyme is the phosphorylated form of the protein in the first layer. The third layer also consists of a protein which can be phosphorylated twice by the same enzyme. This enzyme is the doubly phosphorylated form of the protein in the second layer. The protein in the third layer is the one which is called MAP kinase (mitogen activated protein kinase, MAPK). A kinase is an enzyme which phosphorylates something else and so it is not suprising that the protein in the second layer is called a MAP kinase kinase (MAPKK). The protein in the first layer is accordingly called a MAP kinase kinase kinase (MAPKKK). The roles of the players in this scheme can be taken by different enzymes. For concreteness I name those which occur in the case of human T cells. There the proteins in the first, second and third layers are called Raf, MEK and ERK, respectively. The protein which phophorylates Raf, and hence starts the whole cascade, is Ras. It, or rather the corresponding gene ras, is famous as an oncogene. This means that when the gene is not working properly cancer can result. In fact many drugs used in cancer treatment target proteins belonging to the MAP kinase cascade.
A model for the MAP kinase cascade was written down by Huang and Ferrell (PNAS, 93, 10078). They used a description of Michaelis-Menten type where for each basic substance three species are included in the network. These are the substance itself (free substrate), the enzyme and the complex of the two. Of course since in the MAP kinase cascade certain proteins act both as substrate and enzyme in different reactions there is some overlap between these. For clarity this may be called the ‘extended Michaelis-Menten’ description to contrast it with the ‘effective Michaelis-Menten’ description arising from the extended version by a quasi-steady state limiting process. Note that for a given basic reaction network with species the extended MM description has more than species but still has mass-action kinetics whereas the effective MM description has species but kinetics more complicated than mass action. Phosphatases catalysing the reverse reactions are also included in the model. The phosphatase which removes both phosphate groups of ERK is called MKP3.
In the paper the steady states of the model are studied and an input-output relation is computed numerically. The activity of the MAPK is plotted as a function of the concentration of the first enzyme (Ras in the example). A sigmoidal curve is found which corresponds to what is called ultrasensitivity. The dynamical properties of the model are not discussed. In particular it is not discussed whether there might be multistability (more than one stable stationary solution for fixed values of the parameters) or periodic solutions. The authors also did experiments whose results agreed well with the theoretical predictions. The experiments were done with extracts from the oocytes (immature egg cells) of the frog Xenopus laevis.
The possible dynamic behaviour was investigated in later papers. In some of these the effect of adding an additional feedback was considered. This kind of feedback is probably important in real biological systems. It may, for instance, explain why the results of experiments on whole oocytes are different from those done with extracts. Here, for mathematical simplicity, I will restrict to the case without additional feedback, in other words to the original Huang-Ferrell model. Multistability in this type of model was found in a paper of Markevich, Hoek and Kholodenko (J. Cell Biol. 164, 353). They investigate both extended and effective MM dynamics numerically and find bistability for both. In the extended MM model, which is the one I am most interested in here, the phosphorylation is supposed to be distributive. In other the words the kinase is released between the two phosphorylation steps. The alternative to this is called processive phosphorylation. In a paper of Conradi et. al. this result is compared with chemical reaction network theory (CRNT). It is found that while techniques from CRNT yield results agreeing with those of Markevich et. al. for the case where both the kinase and the phosphatase act in a distributive way, if one of these is replaced by a processive mechanism it can be proved using the Deficiency One Algorithm of CRNT that there is no multistationarity. The case with distributive phosphorylation is the special case of what is called a multiple futile cycle with steps. Wang and Sontag (J. Math Biol. 57, 29) proved upper and lower bounds for the number of steady states in this type of system under certain assumptions on the parameters. In particular this confirms that there can be three steady states (without determining their stability). Going beyond the single layer to the full cascade opens up more possibilities. Numerical evidence has been presented by Qiao et. al. (PLOS Comp. Biol. 9, 2007) that there are periodic solutions. To understand why these should exist it might be best to think of them as relaxation oscillations.